| L(s) = 1 | + 2.05·3-s + 1.64·5-s − 7-s + 1.24·9-s + 1.46·11-s + 3.38·15-s + 1.88·17-s − 6.90·19-s − 2.05·21-s − 0.533·23-s − 2.29·25-s − 3.62·27-s − 5.48·29-s − 10.3·31-s + 3.01·33-s − 1.64·35-s − 1.27·37-s + 0.723·41-s − 2.04·43-s + 2.04·45-s − 0.882·47-s + 49-s + 3.88·51-s − 7.05·53-s + 2.40·55-s − 14.2·57-s + 2.76·59-s + ⋯ |
| L(s) = 1 | + 1.18·3-s + 0.735·5-s − 0.377·7-s + 0.413·9-s + 0.440·11-s + 0.874·15-s + 0.457·17-s − 1.58·19-s − 0.449·21-s − 0.111·23-s − 0.458·25-s − 0.697·27-s − 1.01·29-s − 1.86·31-s + 0.524·33-s − 0.278·35-s − 0.210·37-s + 0.113·41-s − 0.312·43-s + 0.304·45-s − 0.128·47-s + 0.142·49-s + 0.543·51-s − 0.969·53-s + 0.324·55-s − 1.88·57-s + 0.360·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.05T + 3T^{2} \) |
| 5 | \( 1 - 1.64T + 5T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 17 | \( 1 - 1.88T + 17T^{2} \) |
| 19 | \( 1 + 6.90T + 19T^{2} \) |
| 23 | \( 1 + 0.533T + 23T^{2} \) |
| 29 | \( 1 + 5.48T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 1.27T + 37T^{2} \) |
| 41 | \( 1 - 0.723T + 41T^{2} \) |
| 43 | \( 1 + 2.04T + 43T^{2} \) |
| 47 | \( 1 + 0.882T + 47T^{2} \) |
| 53 | \( 1 + 7.05T + 53T^{2} \) |
| 59 | \( 1 - 2.76T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 7.47T + 71T^{2} \) |
| 73 | \( 1 + 1.90T + 73T^{2} \) |
| 79 | \( 1 + 1.00T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 1.43T + 89T^{2} \) |
| 97 | \( 1 + 2.45T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.41801913262968175421876297600, −6.76464253747390085283647392512, −5.92431715904776284433430158470, −5.50688968075591535651884434937, −4.30915216977871630155564833587, −3.71424317170690191524697098846, −3.03076786010233304524586550696, −2.08900811377256001641380483308, −1.70000973442663244855257431460, 0,
1.70000973442663244855257431460, 2.08900811377256001641380483308, 3.03076786010233304524586550696, 3.71424317170690191524697098846, 4.30915216977871630155564833587, 5.50688968075591535651884434937, 5.92431715904776284433430158470, 6.76464253747390085283647392512, 7.41801913262968175421876297600
